is a "pole of order n" for f(z) if and only if the Laurent series for f(z) around has [tex]z^{-n} with non-zero coefficient but no power lower than -n.

But the crucial point here is that cot(z) and coth(z) are analytic at z= 3 while is a rational function with non-zero denominator at z= 3 and so also analytic there. All three functions have Taylor's series (Laurent series with no negative powers) about z= 3 and so there product has a Taylor's series there.

A more interesting problem would be to show that has a pole of order 3 at z= 0.